POSIX sed, 245
==============

POSIX sed, no extensions or extended regexps.  Input is limited to the maximum hold space size of sed - POSIX mandates at least 8192; GNU manages more.  This version assumes that there will be no blank lines before or after the shape; an extra 10 bytes of code, indicated in the expansion, can accommodate that if it's a requirement (original question doesn't specify).

    H
    /^\([L\\]_*\/\|V\| \)*$/!d
    x
    s/[_ ]/  /g
    s/^/!/
    s/$/!/
    :g
    s/\([^V]\)V/V\1/
    tg
    y/V/ /
    s/L/!  /g
    s,[/\\], ! ,g
    s/![^!]*!//g
    :d
    / /{
    s/     /v/g
    s/vv/x/g
    /[ v]/!s/\b/0/2
    s/  /b/g
    s/bb/4/g
    s/b /3/
    s/v /6/
    s/vb/7/
    s/v3/8/
    s/v4/9/
    y/ bvx/125 /
    td
    }

Expanded and annotated
----------------------

    #!/bin/sed -f

    # If leading blank lines may exist, then delete them (and add 8 bytes)
    #/^ *$/d

    # Collect input into hold space until we reach the end of the figure
    H
    /^\([L\\]_*\/\|V\| \)*$/!d

    x

    # Space and underscore each count as two units
    s/[_ ]/  /g

    # Add an edge at the beginning and end, so we can delete matching pairs
    s/^/!/
    s/$/!/
    # Move all the V's to the beginning and convert to a single unit of area
    :gather
    s/\([^V]\)V/V\1/
    tgather
    y/V/ /

    # L is a boundary to left of cell; / and \ in middle
    s/L/!  /g
    s,[/\\], ! ,g

    # Strip out all the bits of outer region
    s/![^!]*!//g

    # Now, we have a space for each unit of area, and no other characters
    # remaining (spaces are convenient because we will use \b to match
    # where they end).  To count the spaces, we use roman numerals v and x
    # to match five and ten, respectively.  We also match two (and call
    # that 'b').  At the end of the loop, tens are turned back into spaces
    # again.
    :digit
    / /{
    s/     /v/g
    s/vv/x/g
    /[ v]/!s/\b/0/2
    s/  /b/g
    s/bb/4/g
    s/b /3/
    s/v /6/
    s/vb/7/
    s/v3/8/
    s/v4/9/
    y/ bvx/125 /
    tdigit
    }

    # If trailing blank lines may exist, then stop now (and add 2 bytes)
    #q